Intuitively the difference between to elements in S is the biggest when one is the biggest element while the other one is the smallest (or vice versa).

Jan 14th 2011, 04:30 AM

luckylawrance

Respected, the idea i have is that difference between two numbers is greatest when one is largest and other is smallest in S. but i don't know how to write the proof in proper way.

Jan 15th 2011, 03:54 PM

Drexel28

Quote:

Originally Posted by luckylawrance

Respected, the idea i have is that difference between two numbers is greatest when one is largest and other is smallest in S. but i don't know how to write the proof in proper way.

Use the characterization that if $\displaystyle A\subseteq\mathbb{R}$ is bounded then $\displaystyle \sup A=\alpha$ if and only if $\displaystyle \alpha$ is an upper bound of $\displaystyle A$ and for every $\displaystyle \varepsilon>0$ there exists $\displaystyle a\in A$ such that $\displaystyle \alpha-\varepsilon<\alpha$.